How to Graph Linear Equations — Slope-Intercept and Standard Form

Grade: 8–9 | Topic: Algebra

What You Will Learn

By the end of this page, you will be able to graph a linear equation using slope-intercept form ($y = mx + b$), using intercepts, and from a table of values. You will also understand what the slope and y-intercept tell you about how the graph looks before you even draw it.

Theory

Slope-Intercept Form: $y = mx + b$

Every straight line can be written as $y = mx + b$, where:

• $m$ = slope (how steep the line is and which direction)
• $b$ = y-intercept (where the line crosses the y-axis)

To graph from slope-intercept form:

1. Plot the y-intercept: the point $(0, b)$
2. Use the slope to find a second point: from $(0, b)$, move right by the denominator of $m$ and up/down by the numerator
3. Draw a straight line through both points and extend it with arrows

Finding Intercepts from Standard Form $Ax + By = C$

To find the y-intercept: set $x = 0$ and solve for $y$.

To find the x-intercept: set $y = 0$ and solve for $x$.

Plot both intercepts, then draw the line.

Graphing from a Table of Values

Create a table by substituting values of $x$ into the equation to get $y$. Plot the resulting coordinate pairs and connect them.

Worked Examples

Example 1: Graphing from Slope-Intercept Form

Problem: Graph $y = 2x - 3$.

Step 1: Identify slope and y-intercept. $m = 2 = \frac{2}{1}, \quad b = -3$

Step 2: Plot the y-intercept $(0, -3)$.

Step 3: Use the slope. From $(0, -3)$, move 1 unit right and 2 units up to reach $(1, -1)$. Repeat to get $(2, 1)$.

Step 4: Draw a line through the points with arrows on both ends.

Answer: A line through $(0, -3)$, $(1, -1)$, $(2, 1)$, rising steeply from left to right.

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Example 2: Graphing with a Fractional Slope

Problem: Graph $y = -\dfrac{1}{3}x + 4$.

Step 1: $m = -\dfrac{1}{3}$ (fall 1, run 3), $b = 4$.

Step 2: Plot $(0, 4)$.

Step 3: From $(0, 4)$, move 3 right and 1 down to reach $(3, 3)$. Continue to $(6, 2)$.

Answer: A gently sloping line through $(0, 4)$, $(3, 3)$, $(6, 2)$, falling from left to right.

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Example 3: Graphing Using Intercepts (Standard Form)

Problem: Graph $3x + 4y = 12$.

Step 1: Find the y-intercept (set $x = 0$). $3(0) + 4y = 12 \implies y = 3 \implies \text{point: } (0, 3)$

Step 2: Find the x-intercept (set $y = 0$). $3x + 4(0) = 12 \implies x = 4 \implies \text{point: } (4, 0)$

Step 3: Plot $(0, 3)$ and $(4, 0)$, then draw a line through them.

Answer: A line from $(0, 3)$ to $(4, 0)$, falling left to right.

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Example 4: Graphing from a Table

Problem: Graph the equation $y = -x + 5$ using a table.

Step 1: Choose values for $x$ and calculate $y$.

$x$	$-1$	$0$	$1$	$2$	$3$
$y$	$6$	$5$	$4$	$3$	$2$

Step 2: Plot the five points and draw a straight line through them.

Answer: A downward-sloping line through these points.

Common Mistakes

Mistake 1: Plotting Slope as (run, rise) Instead of (rise, run)

❌ For slope $m = \frac{3}{4}$, student moves 3 right and 4 up (treating it as run/rise).

✅ Move 4 right (run) and 3 up (rise). The numerator is rise ($\Delta y$), denominator is run ($\Delta x$).

Mistake 2: Forgetting the Negative Sign in a Negative Slope

❌ $m = -\frac{2}{3}$: student plots by going 2 units right and 2 units up.

✅ Go 3 right and 2 down (because the slope is negative — the line falls).

Mistake 3: Stopping After Two Points

❌ Drawing a line segment just between the two plotted points.

✅ A line extends in both directions infinitely. Always draw arrows at both ends.

Practice Problems

Try these on your own before checking the answers:

1. Identify the slope and y-intercept of $y = -4x + 7$.
2. Graph $y = \dfrac{1}{2}x - 1$ (describe the y-intercept and a second point).
3. Find both intercepts of $2x - 5y = 10$.
4. Does the point $(3, 8)$ lie on the line $y = 2x + 2$?
5. Write the equation of a line with slope $-3$ and y-intercept $5$.

Click to see answers

1. Slope $= -4$, y-intercept $= 7$ (point $(0, 7)$)
2. y-intercept: $(0, -1)$. Move 2 right, 1 up to get $(2, 0)$.
3. y-intercept: set $x = 0$: $-5y = 10 \implies y = -2$, so $(0, -2)$. x-intercept: set $y = 0$: $2x = 10 \implies x = 5$, so $(5, 0)$.
4. $y = 2(3) + 2 = 8$. Yes, $(3, 8)$ lies on the line.
5. $y = -3x + 5$

Summary

• Slope-intercept method: plot $(0, b)$, then use slope $m = \frac{\text{rise}}{\text{run}}$ to find a second point.
• Intercept method: find $(0, y\text{-intercept})$ and $(x\text{-intercept}, 0)$ by substituting $x = 0$ and $y = 0$.
• Positive slope = line goes up left to right; negative slope = goes down.
• Always extend the line in both directions with arrows.

Related Topics

• How to Find the Slope of a Line
• Linear Equations — How to Solve Step by Step
• The Coordinate Plane — Plotting Points

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